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Heuristic Search

Heuristic Search is a search algorithm that uses heuristic functions to guide the search process. The heuristic function estimates the cost to reach the goal from a given state, allowing the algorithm to prioritize which paths to explore. This approach is commonly used in AI and optimization problems.

Algorithm Steps

  1. Initialization:

    • Define the heuristic function that estimates the cost to reach the goal.
    • Initialize the search from the starting state.
  2. Search Process:

    • Use a priority queue to manage the states to be explored, prioritizing states with lower heuristic costs.
    • At each step, expand the state with the lowest estimated cost.
    • If the goal state is reached, return the path or the solution.

Complexity Analysis

  • Time Complexity: Depends on the heuristic function. In the best case, it can be O(logn)O(\log n), but in the worst case, it can approach O(n)O(n) where nn is the number of states.
  • Space Complexity: O(n)O(n) for storing the priority queue and explored states.

Example

Given a graph representing a map:

graph = {
'A': [('B', 1), ('C', 3)],
'B': [('D', 5)],
'C': [('D', 2)],
'D': []
}
start = 'A'

goal = 'D'

Using Heuristic Search:

  • Define a heuristic function that estimates the distance from each state to the goal.
  • Use the heuristic to guide the search from the start state to the goal state.

Implementation

import heapq

def heuristic(node, goal):
# Example heuristic: distance to goal
heuristics = {
'A': 4,
'B': 2,
'C': 2,
'D': 0
}
return heuristics[node]

def heuristic_search(graph, start, goal):
priority_queue = [(0 + heuristic(start, goal), start)]
explored = set()

while priority_queue:
cost, current = heapq.heappop(priority_queue)
if current == goal:
return cost

if current not in explored:
explored.add(current)
for neighbor, weight in graph[current]:
if neighbor not in explored:
total_cost = cost + weight
heapq.heappush(priority_queue, (total_cost + heuristic(neighbor, goal), neighbor))

return float('inf')

# Example usage:
graph = {
'A': [('B', 1), ('C', 3)],
'B': [('D', 5)],
'C': [('D', 2)],
'D': []
}
start = 'A'
goal = 'D'

result = heuristic_search(graph, start, goal)
print(f"Minimum cost from {start} to {goal}: {result}")

Conclusion

The Heuristic Search algorithm efficiently searches for a target state by leveraging heuristic functions to guide the search process. It is widely used in AI, optimization problems, and various applications where informed search strategies are beneficial.